CBSE Class 10 Maths: Polynomials — Zeroes & Division Notes 2026

Types of Polynomials

• Degree 1: linear (ax+b); degree 2: quadratic (ax²+bx+c); degree 3: cubic
• Zeroes of polynomial: values of x where p(x)=0; linear has 1 zero; quadratic has up to 2
• Graph of linear: straight line; quadratic: parabola; zeroes are x-intercepts

Zeroes and Coefficients (Quadratic)

• For ax² + bx + c = 0 with roots α and β:
• Sum of zeroes α + β = −b/a; product αβ = c/a
• Form quadratic given its zeroes: x² − (sum)x + (product) = 0

Zeroes and Coefficients (Cubic)

• For ax³ + bx² + cx + d with zeroes α, β, γ:
• α+β+γ = −b/a; αβ+βγ+γα = c/a; αβγ = −d/a
• CBSE: verify relationship given zeroes; find third zero given two

Graphical Understanding of Zeroes

• Zeroes = x-coordinates where graph crosses/touches x-axis
• Quadratic with no real zeroes: parabola entirely above or below x-axis (D<0)
• Cubic polynomial always has at least one real zero (odd degree)

Division Algorithm for Polynomials

• If p(x) is divided by g(x): p(x) = g(x)·q(x) + r(x); deg(r) < deg(g) or r=0
• Application: find q(x) and r(x); verify by multiplication
• Find all zeroes of cubic if one zero is given; use division to reduce to quadratic

Finding Zeroes of Polynomials

• Quadratic: factorisation or quadratic formula
• Cubic: if one zero k is given, (x−k) is a factor; divide to get quadratic, then solve
• CBSE standard question: given p(x) = 2x³−5x²−14x+8 with zero ½; find all zeroes

CBSE Exam Tips — Polynomials

• Memorise sum/product formulas for quadratic and cubic exactly
• Always verify zeroes are correct by substitution before writing final answer
• Common error: forgetting that coefficient of x² must be made 1 in sum/product formula